{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T22:05:02Z","timestamp":1774044302833,"version":"3.50.1"},"reference-count":37,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871400"],"award-info":[{"award-number":["11871400"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2025YFA1016800"],"award-info":[{"award-number":["2025YFA1016800"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.cam.2026.117496","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T07:45:04Z","timestamp":1772610304000},"page":"117496","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Riemannian optimization for model order reduction of linear systems with quadratic outputs"],"prefix":"10.1016","volume":"485","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5573-2047","authenticated-orcid":false,"given":"Xiaolong","family":"Wang","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0005-2076-1215","authenticated-orcid":false,"given":"Tongtu","family":"Tian","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2026.117496_bib0001","series-title":"Model Reduction and Approximation: Theory and Algorithms","author":"Benner","year":"2017"},{"key":"10.1016\/j.cam.2026.117496_bib0002","series-title":"Interpolatory Methods for Model Reduction","author":"Antoulas","year":"2020"},{"issue":"125","key":"10.1016\/j.cam.2026.117496_bib0003","article-title":"Identification of dominant subspaces for model reduction of structured parametric systems","volume":"15","author":"Benner","year":"2024","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"3","key":"10.1016\/j.cam.2026.117496_bib0004","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.sysconle.2008.10.016","article-title":"Interpolatory projection methods for structure-preserving model reduction","volume":"58","author":"Beattie","year":"2009","journal-title":"Syst. Control Lett."},{"issue":"692","key":"10.1016\/j.cam.2026.117496_bib0005","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.laa.2024.04.004","article-title":"Model order reduction for discrete time-delay systems based on Laguerre function expansion","author":"Wang","year":"2024","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"10.1016\/j.cam.2026.117496_bib0006","first-page":"2033","article-title":"Model reduction by balanced truncation of linear systems with a quadratic output","volume":"3","author":"Van Beeumen","year":"2010","journal-title":"Am. Inst. Phys."},{"issue":"3","key":"10.1016\/j.cam.2026.117496_bib0007","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1002\/nme.4255","article-title":"Model reduction for dynamical systems with quadratic output","volume":"91","author":"Van Beeumen","year":"2012","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"4","key":"10.1016\/j.cam.2026.117496_bib0008","doi-asserted-by":"crossref","first-page":"A2270","DOI":"10.1137\/17M1148797","article-title":"Balanced truncation for model order reduction of linear dynamical systems with quadratic outputs","volume":"41","author":"Pulch","year":"2019","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"10.1016\/j.cam.2026.117496_bib0009","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1109\/TAC.2021.3086319","article-title":"Gramians, energy functionals, and balanced truncation for linear dynamical systems with quadratic outputs","volume":"67","author":"Benner","year":"2021","journal-title":"IEEE Trans. Autom. Control"},{"key":"10.1016\/j.cam.2026.117496_bib0010","series-title":"2019 IEEE 58th Conference on Decision and Control (CDC)","first-page":"7812","article-title":"A two-sided iterative framework for model reduction of linear systems with quadratic output","author":"Gosea","year":"2019"},{"issue":"1","key":"10.1016\/j.cam.2026.117496_bib0011","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s10915-022-01771-5","article-title":"Data-driven modeling of linear dynamical systems with quadratic output in the AAA framework","volume":"91","author":"Gosea","year":"2022","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.cam.2026.117496_bib0012","doi-asserted-by":"crossref","unstructured":"S. Reiter, I.P. Duff, I.V. Gosea, S. Gugercin, H2 optimal model reduction of linear systems with multiple quadratic outputs, https:\/\/arxiv.org\/abs\/2405.05951v1(2024) 1\u201318.","DOI":"10.1109\/TAC.2025.3636441"},{"issue":"2","key":"10.1016\/j.cam.2026.117496_bib0013","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1137\/060666123","article-title":"H2 model reduction for large-scale linear dynamical systems","volume":"30","author":"Gugercin","year":"2008","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"14","key":"10.1016\/j.cam.2026.117496_bib0014","first-page":"1341","article-title":"An approximate approach to H2 optimal model reduction","volume":"7","author":"Yan","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"issue":"10","key":"10.1016\/j.cam.2026.117496_bib0015","first-page":"972","article-title":"Fast optimal H2 model reduction algorithms based on Grassmann manifold optimization","volume":"4","author":"Xu","year":"2013","journal-title":"Int. J. Numer. Anal. Model."},{"issue":"3","key":"10.1016\/j.cam.2026.117496_bib0016","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1080\/00207179.2018.1482501","article-title":"H2 model reduction for negative imaginary systems","volume":"93","author":"Yu","year":"2020","journal-title":"Int. J. Control"},{"issue":"1","key":"10.1016\/j.cam.2026.117496_bib0017","first-page":"2","article-title":"Riemannian optimal model reduction of linear second-order systems","volume":"1","author":"Sato","year":"2017","journal-title":"IEEE Control Syst. Lett."},{"issue":"62","key":"10.1016\/j.cam.2026.117496_bib0018","doi-asserted-by":"crossref","first-page":"6575","DOI":"10.1109\/TAC.2017.2712905","article-title":"Riemannian optimal control and model matching of linear port-Hamiltonian systems","volume":"12","author":"Sato","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"issue":"10","key":"10.1016\/j.cam.2026.117496_bib0019","doi-asserted-by":"crossref","first-page":"4467","DOI":"10.1016\/j.jfranklin.2015.06.022","article-title":"H2 order-reduction for bilinear systems based on Grassmann manifold","volume":"352","author":"Xu","year":"2015","journal-title":"J. Frankl. Inst."},{"issue":"41","key":"10.1016\/j.cam.2026.117496_bib0020","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1137\/19M1257147","article-title":"Riemannian manifold Polak-Ribiere-Polyak conjugate gradient order reduced model by tensor techniques","volume":"2","author":"Jiang","year":"2020","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"52","key":"10.1016\/j.cam.2026.117496_bib0021","doi-asserted-by":"crossref","first-page":"1928","DOI":"10.1137\/130927723","article-title":"An online manifold learning approach for model reduction of dynamical systems","volume":"4","author":"Peng","year":"2014","journal-title":"SIAM J. Numer. Anal."},{"issue":"68","key":"10.1016\/j.cam.2026.117496_bib0022","doi-asserted-by":"crossref","first-page":"5616","DOI":"10.1109\/TAC.2022.3218620","article-title":"Reduced model reconstruction method for stable positive network systems","volume":"9","author":"Sato","year":"2023","journal-title":"IEEE Trans. Autom. Control"},{"issue":"5","key":"10.1016\/j.cam.2026.117496_bib0023","doi-asserted-by":"crossref","first-page":"2979","DOI":"10.1109\/TAC.2024.3489416","article-title":"IRKA Is a Riemannian gradient descent method","volume":"70","author":"Mlinaric","year":"2025","journal-title":"IEEE Trans. Autom. Control"},{"issue":"2","key":"10.1016\/j.cam.2026.117496_bib0024","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1137\/11082885X","article-title":"Optimization methods on Riemannian manifolds and their application to shape space","volume":"22","author":"Ring","year":"2012","journal-title":"SIAM J. Optim."},{"key":"10.1016\/j.cam.2026.117496_bib0025","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s10589-015-9801-1","article-title":"A Dai\u2013Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions","volume":"64","author":"Sato","year":"2016","journal-title":"Comput. Optim. Appl."},{"issue":"28","key":"10.1016\/j.cam.2026.117496_bib0026","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1137\/17M1127582","article-title":"A Riemannian BFGS method without differentiated retraction for nonconvex optimization problems","volume":"1","author":"Huang","year":"2018","journal-title":"SIAM J. Optim."},{"key":"10.1016\/j.cam.2026.117496_bib0027","series-title":"Optimization Algorithms on Matrix Manifolds","author":"Absil","year":"2008"},{"key":"10.1016\/j.cam.2026.117496_bib0028","series-title":"An Introduction to Optimization on Smooth Manifolds","author":"Boumal","year":"2023"},{"issue":"3","key":"10.1016\/j.cam.2026.117496_bib0029","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1137\/130912839","article-title":"Computational methods for linear matrix equations","volume":"58","author":"Simoncini","year":"2016","journal-title":"SIAM Rev."},{"key":"10.1016\/j.cam.2026.117496_bib0030","series-title":"Theoretical Numerical Analysis","author":"Atkinson","year":"2005"},{"issue":"9","key":"10.1016\/j.cam.2026.117496_bib0031","doi-asserted-by":"crossref","first-page":"1469","DOI":"10.1109\/22.868996","article-title":"Laguerre-SVD reduced-order modeling","volume":"48","author":"Knockaert","year":"2000","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"10.1016\/j.cam.2026.117496_bib0032","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.apm.2022.01.035","article-title":"Model order reduction of linear and bilinear systems via low-rank Gramian approximation","volume":"106","author":"Xiao","year":"2022","journal-title":"Appl. Math. Model."},{"issue":"3","key":"10.1016\/j.cam.2026.117496_bib0033","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1016\/j.laa.2010.09.021","article-title":"Orthogonal polynomial expansions for the matrix exponential","volume":"435","author":"Moore","year":"2011","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"10.1016\/j.cam.2026.117496_bib0034","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1016\/j.jfranklin.2012.01.009","article-title":"Model reduction of bilinear systems based on Laguerre series expansion","volume":"24","author":"Wang","year":"2012","journal-title":"J. Frankl. Inst."},{"issue":"1","key":"10.1016\/j.cam.2026.117496_bib0035","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1137\/S0895479801384937","article-title":"Low rank solution of Lyapunov equations","volume":"24","author":"Li","year":"2002","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"10.1016\/j.cam.2026.117496_bib0036","series-title":"2016 IEEE 55th Conference on Decision and Control (CDC)","first-page":"5762","article-title":"A new H2 optimal model reduction method based on Riemannian conjugate gradient method","author":"Sato","year":"2016"},{"key":"10.1016\/j.cam.2026.117496_bib0037","unstructured":"Y. Chahlaoui, P.V. Dooren, A collection of benchmark examples for model reduction of linear time invariant dynamical systems, http:\/\/eprints.maths.manchester.ac.uk\/1040\/1\/ChahlaouiV02a.pdf.(2002)."}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726001597?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726001597?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T20:21:02Z","timestamp":1774038062000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042726001597"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":37,"alternative-id":["S0377042726001597"],"URL":"https:\/\/doi.org\/10.1016\/j.cam.2026.117496","relation":{},"ISSN":["0377-0427"],"issn-type":[{"value":"0377-0427","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Riemannian optimization for model order reduction of linear systems with quadratic outputs","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cam.2026.117496","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"117496"}}